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  <div class="section" id="numpy-fft-irfft">
<h1>numpy.fft.irfft<a class="headerlink" href="#numpy-fft-irfft" title="Permalink to this headline">¶</a></h1>
<dl class="function">
<dt id="numpy.fft.irfft">
<code class="sig-prename descclassname">numpy.fft.</code><code class="sig-name descname">irfft</code><span class="sig-paren">(</span><em class="sig-param">a</em>, <em class="sig-param">n=None</em>, <em class="sig-param">axis=-1</em>, <em class="sig-param">norm=None</em><span class="sig-paren">)</span><a class="reference external" href="https://github.com/numpy/numpy/blob/v1.18.1/numpy/fft/_pocketfft.py#L378-L475"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#numpy.fft.irfft" title="Permalink to this definition">¶</a></dt>
<dd><p>Compute the inverse of the n-point DFT for real input.</p>
<p>This function computes the inverse of the one-dimensional <em>n</em>-point
discrete Fourier Transform of real input computed by <a class="reference internal" href="numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a>.
In other words, <code class="docutils literal notranslate"><span class="pre">irfft(rfft(a),</span> <span class="pre">len(a))</span> <span class="pre">==</span> <span class="pre">a</span></code> to within numerical
accuracy. (See Notes below for why <code class="docutils literal notranslate"><span class="pre">len(a)</span></code> is necessary here.)</p>
<p>The input is expected to be in the form returned by <a class="reference internal" href="numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a>, i.e. the
real zero-frequency term followed by the complex positive frequency terms
in order of increasing frequency.  Since the discrete Fourier Transform of
real input is Hermitian-symmetric, the negative frequency terms are taken
to be the complex conjugates of the corresponding positive frequency terms.</p>
<dl class="field-list">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><dl>
<dt><strong>a</strong><span class="classifier">array_like</span></dt><dd><p>The input array.</p>
</dd>
<dt><strong>n</strong><span class="classifier">int, optional</span></dt><dd><p>Length of the transformed axis of the output.
For <em class="xref py py-obj">n</em> output points, <code class="docutils literal notranslate"><span class="pre">n//2+1</span></code> input points are necessary.  If the
input is longer than this, it is cropped.  If it is shorter than this,
it is padded with zeros.  If <em class="xref py py-obj">n</em> is not given, it is taken to be
<code class="docutils literal notranslate"><span class="pre">2*(m-1)</span></code> where <code class="docutils literal notranslate"><span class="pre">m</span></code> is the length of the input along the axis
specified by <em class="xref py py-obj">axis</em>.</p>
</dd>
<dt><strong>axis</strong><span class="classifier">int, optional</span></dt><dd><p>Axis over which to compute the inverse FFT. If not given, the last
axis is used.</p>
</dd>
<dt><strong>norm</strong><span class="classifier">{None, “ortho”}, optional</span></dt><dd><div class="versionadded">
<p><span class="versionmodified added">New in version 1.10.0.</span></p>
</div>
<p>Normalization mode (see <a class="reference internal" href="../routines.fft.html#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a>). Default is None.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><dl class="simple">
<dt><strong>out</strong><span class="classifier">ndarray</span></dt><dd><p>The truncated or zero-padded input, transformed along the axis
indicated by <em class="xref py py-obj">axis</em>, or the last one if <em class="xref py py-obj">axis</em> is not specified.
The length of the transformed axis is <em class="xref py py-obj">n</em>, or, if <em class="xref py py-obj">n</em> is not given,
<code class="docutils literal notranslate"><span class="pre">2*(m-1)</span></code> where <code class="docutils literal notranslate"><span class="pre">m</span></code> is the length of the transformed axis of the
input. To get an odd number of output points, <em class="xref py py-obj">n</em> must be specified.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises</dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>IndexError</strong></dt><dd><p>If <em class="xref py py-obj">axis</em> is larger than the last axis of <em class="xref py py-obj">a</em>.</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="../routines.fft.html#module-numpy.fft" title="numpy.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">numpy.fft</span></code></a></dt><dd><p>For definition of the DFT and conventions used.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.rfft.html#numpy.fft.rfft" title="numpy.fft.rfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">rfft</span></code></a></dt><dd><p>The one-dimensional FFT of real input, of which <a class="reference internal" href="#numpy.fft.irfft" title="numpy.fft.irfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft</span></code></a> is inverse.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.fft.html#numpy.fft.fft" title="numpy.fft.fft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">fft</span></code></a></dt><dd><p>The one-dimensional FFT.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.irfft2.html#numpy.fft.irfft2" title="numpy.fft.irfft2"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft2</span></code></a></dt><dd><p>The inverse of the two-dimensional FFT of real input.</p>
</dd>
<dt><a class="reference internal" href="numpy.fft.irfftn.html#numpy.fft.irfftn" title="numpy.fft.irfftn"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfftn</span></code></a></dt><dd><p>The inverse of the <em>n</em>-dimensional FFT of real input.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>Returns the real valued <em class="xref py py-obj">n</em>-point inverse discrete Fourier transform
of <em class="xref py py-obj">a</em>, where <em class="xref py py-obj">a</em> contains the non-negative frequency terms of a
Hermitian-symmetric sequence. <em class="xref py py-obj">n</em> is the length of the result, not the
input.</p>
<p>If you specify an <em class="xref py py-obj">n</em> such that <em class="xref py py-obj">a</em> must be zero-padded or truncated, the
extra/removed values will be added/removed at high frequencies. One can
thus resample a series to <em class="xref py py-obj">m</em> points via Fourier interpolation by:
<code class="docutils literal notranslate"><span class="pre">a_resamp</span> <span class="pre">=</span> <span class="pre">irfft(rfft(a),</span> <span class="pre">m)</span></code>.</p>
<p>The correct interpretation of the hermitian input depends on the length of
the original data, as given by <em class="xref py py-obj">n</em>. This is because each input shape could
correspond to either an odd or even length signal. By default, <a class="reference internal" href="#numpy.fft.irfft" title="numpy.fft.irfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft</span></code></a>
assumes an even output length which puts the last entry at the Nyquist
frequency; aliasing with its symmetric counterpart. By Hermitian symmetry,
the value is thus treated as purely real. To avoid losing information, the
correct length of the real input <strong>must</strong> be given.</p>
<p class="rubric">Examples</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">ifft</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="n">j</span><span class="p">])</span>
<span class="go">array([0.+0.j,  1.+0.j,  0.+0.j,  0.+0.j]) # may vary</span>
<span class="gp">&gt;&gt;&gt; </span><span class="n">np</span><span class="o">.</span><span class="n">fft</span><span class="o">.</span><span class="n">irfft</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="n">j</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">])</span>
<span class="go">array([0.,  1.,  0.,  0.])</span>
</pre></div>
</div>
<p>Notice how the last term in the input to the ordinary <a class="reference internal" href="numpy.fft.ifft.html#numpy.fft.ifft" title="numpy.fft.ifft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">ifft</span></code></a> is the
complex conjugate of the second term, and the output has zero imaginary
part everywhere.  When calling <a class="reference internal" href="#numpy.fft.irfft" title="numpy.fft.irfft"><code class="xref py py-obj docutils literal notranslate"><span class="pre">irfft</span></code></a>, the negative frequencies are not
specified, and the output array is purely real.</p>
</dd></dl>

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